In the semi-classical theory of NMR relaxation, correlation functions are treated classically and their ensemble averages use classical weighting. In the quantum mechanical formulation for rotationally symmetric groups, there are restrictions on the symmetry of the total wave function, i.e. the product of nuclear spin, rotation, vibration, and electronic wave functions. In addition, the interaction that produces NMR relaxation is a product of nuclear spin and rotational operators. The symmetrically correct wave functions combine with the operators to produce quantum statistical weights for the correlation functions that can differ from classical weights. The quantum statistical weights for quadrupole relaxation of deuterium in a CD3 group and for dipole-dipole relaxation of A3 and AX3 systems of spin 1/2 nuclei have been derived. The weights for deuterium relaxation are found to be the same as classical weights. The dipolar relaxation, on the other hand, has eight different weighting schemes for different types of correlation functions, one of which is the classical weight. In addition, certain terms in the dipolar relaxation matrix, which are zero in the semi-classical theory, are non-zero in the quantum mechanical treatment. Under certain circumstances, use of the quantum statistical weights leads to predicted relaxation behavior that is significantly different from that found with classical weights. And the theory predicts that the NMR relaxation behavior can change if the symmetry of the product of vibrational and electronic wave functions is changed, e.g. through infrared excitation.